Finite Math Examples

$x - \frac{y}{\frac{x}{y} + \frac{y}{x}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

To write $\frac{x}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$x - \frac{y}{\frac{x}{y} \cdot \frac{x}{x} + \frac{y}{x}}$

Step 1.1.2

To write $\frac{y}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$x - \frac{y}{\frac{x}{y} \cdot \frac{x}{x} + \frac{y}{x} \cdot \frac{y}{y}}$

Step 1.1.3

Write each expression with a common denominator of $y x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.3.1

Multiply $\frac{x}{y}$ by $\frac{x}{x}$ .

$x - \frac{y}{\frac{x \cdot x}{y x} + \frac{y}{x} \cdot \frac{y}{y}}$

Step 1.1.3.2

Multiply $\frac{y}{x}$ by $\frac{y}{y}$ .

$x - \frac{y}{\frac{x \cdot x}{y x} + \frac{y \cdot y}{x y}}$

Step 1.1.3.3

Reorder the factors of $y x$ .

$x - \frac{y}{\frac{x \cdot x}{x y} + \frac{y \cdot y}{x y}}$

Step 1.1.4

Combine the numerators over the common denominator.

$x - \frac{y}{\frac{x \cdot x + y \cdot y}{x y}}$

Step 1.1.5

Rewrite $\frac{x \cdot x + y \cdot y}{x y}$ in a factored form.

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Step 1.1.5.1

Multiply $x$ by $x$ .

$x - \frac{y}{\frac{x^{2} + y \cdot y}{x y}}$

Step 1.1.5.2

Multiply $y$ by $y$ .

$x - \frac{y}{\frac{x^{2} + y^{2}}{x y}}$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$x - (y \frac{x y}{x^{2} + y^{2}})$

Step 1.3

Multiply $y \frac{x y}{x^{2} + y^{2}}$ .

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Step 1.3.1

Combine $y$ and $\frac{x y}{x^{2} + y^{2}}$ .

$x - \frac{y (x y)}{x^{2} + y^{2}}$

Step 1.3.2

Raise $y$ to the power of $1$ .

$x - \frac{x (y^{1} y)}{x^{2} + y^{2}}$

Step 1.3.3

Raise $y$ to the power of $1$ .

$x - \frac{x (y^{1} y^{1})}{x^{2} + y^{2}}$

Step 1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - \frac{x y^{1 + 1}}{x^{2} + y^{2}}$

Step 1.3.5

Add $1$ and $1$ .

$x - \frac{x y^{2}}{x^{2} + y^{2}}$

Step 2

To write $x$ as a fraction with a common denominator, multiply by $\frac{x^{2} + y^{2}}{x^{2} + y^{2}}$ .

$\frac{x (x^{2} + y^{2})}{x^{2} + y^{2}} - \frac{x y^{2}}{x^{2} + y^{2}}$

Step 3

Combine the numerators over the common denominator.

$\frac{x (x^{2} + y^{2}) - x y^{2}}{x^{2} + y^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $x$ out of $x (x^{2} + y^{2}) - x y^{2}$ .

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Step 4.1.1

Factor $x$ out of $- x y^{2}$ .

$\frac{x (x^{2} + y^{2}) + x (- y^{2})}{x^{2} + y^{2}}$

Step 4.1.2

Factor $x$ out of $x (x^{2} + y^{2}) + x (- y^{2})$ .

$\frac{x (x^{2} + y^{2} - y^{2})}{x^{2} + y^{2}}$

Step 4.2

Subtract $y^{2}$ from $y^{2}$ .

$\frac{x (x^{2} + 0)}{x^{2} + y^{2}}$

Step 4.3

Add $x^{2}$ and $0$ .

$\frac{x \cdot x^{2}}{x^{2} + y^{2}}$

Step 5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.1

Multiply $x$ by $x^{2}$ .

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Step 5.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{2}}{x^{2} + y^{2}}$

Step 5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 2}}{x^{2} + y^{2}}$

Step 5.2

Add $1$ and $2$ .

$\frac{x^{3}}{x^{2} + y^{2}}$